Is it possible to extrude a curve in two directions?
Not supported in Blender, curve vertex must always belong to a maximum of two segments.

Either use a mesh object with a Skin modifier instead, if you require true 'branching' and a continuous mesh surface. 



Or if you prefer curve objects duplicate a vertex in place, leave it overlapping the original one, and then extrude the new one; if you don't mind the discontinuity in the mesh.



You can just directly select any middle vertex and extrude it twice with E, the first extrude will duplicate the vertex automatically Right-Click to cancel and the second extrude will start a new branch.